Can a CNN be trained incrementally?
You are looking for incremental (or online) learning.

A CNN can be trained incrementally. For example, in the paper Incremental Learning of Convolutional Neural Networks, the authors propose an incremental learning algorithm (inspired by AdaBoost and Learn++, which is another incremental learning algorithm for supervised learning of neural networks) for CNNs. 

However, note that incremental learning is a challenging task, given the  stability-plasticity dilemma: a completely stable model, in order to keep being stable, will attempt to preserve the existing knowledge, so it will not learn new knowledge; similarly, a completely plastic model, in order to keep being plastic, it will keep forgetting previously acquired knowledge so that to learn new information.